Antigoni Kaliontzopoulou, CIBIO/InBIO, University of Porto
14 October, 2019
Morphometrics: variation in shape and its covariation with other variables
Morphometrics: variation in shape and its covariation with other variables
Position and orientation are due to digitizing procedures, and are not biologically relevant for most applications
Scale includes the combined effects of focal distance during digitizing and “real” size variation
Digitizing scale is calibrated during data acquisition
Size is of biological interest. So, we standardize for it to obtain shape variables, but record it for subsequent analyses
Small objects: landmarks are closer together
Large objects: landmarks are further apart
So, in GM, size is associated to the dispersion of landmark coordinates
Small objects: landmarks are closer together
Large objects: landmarks are further apart
So, in GM, size is associated to the dispersion of landmark coordinates
Centroid size: the square root of the sum of the squared distances between each landmark and the centroid (center of mass of the object) of the landmark configuration:
\[\small{CS}=\sqrt{\sum_{i,j}^{k,p}\left(\mathbf{Y}_{ij}-\mathbf{Y}_{ic}\right)^2}\]
where \(\small{p}\) is the number of landmarks and \(\small{k}\) is the number of coordinate dimensions
\[\small{D}_{Proc}=\sqrt{\sum_{i,j}^{k,p}\left(\mathbf{Y}_{1.ij}-\mathbf{Y}_{2.ij}\right)^2}\]
\(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)
\[\small{CS}=\sqrt{\sum_{i,j}^{k,p}\left(\mathbf{Y}_{ij}-\mathbf{Y}_{ic}\right)^2}=\]
\[\small{[tr}[\mathbf{(Y-\overline{Y})(Y-\overline{Y})^T}]]^{-1/2}=1\]
\(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)
In Procrustes, rotation accomplished by rigid rotation using as: : \(\small\mathbf{H}=\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}\)
\(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)
In Procrustes, rotation is accomplished by rigid rotation using: \(\small\mathbf{H}=\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}\)
SVD as: \(\small\mathbf{H}=\mathbf{VSU}^T\)\(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)
In Procrustes, rotation accomplished by rigid rotation using as: : \(\small\mathbf{H}=\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}\)
SVD as: \(\small\mathbf{H}=\mathbf{VSU}^T\)\(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)
Perform OPA in iterative fashion
Perform OPA in iterative fashion
Sexual dimorphism in head shape of Podarcis wall lizards
12 lateral 2D landmarks
Males and females
MANOVA to test for shape differences
PCA to visualize shape space
## Df SS MS Rsq F Z Pr(>F)
## gp 1 0.009180 0.0091801 0.09059 5.8772 4.5952 0.001 **
## Residuals 59 0.092158 0.0015620 0.90941
## Total 60 0.101338
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## [1] 0.019120833 0.014962558 0.013953399 0.011872076 0.011252644
## [6] 0.010428595 0.010172139 0.008714636 0.008158964 0.007586907
## [11] 0.006831716 0.006049213 0.005992073 0.005275534 0.004900721
## [16] 0.004129789 0.003339428 0.003200212 0.002954954 0.002591265
## [21] 0.000000000 0.000000000 0.000000000 0.000000000
Therefore, the shape space for GPA-aligned specimens has \(\small{2p-4}\) dimensions for 2D data
For 3D data, dimensionality is \(\small{3p-7}\) (general form is: \(\small{pk – k – k(k – 1)/2 – 1}\)
Therefore, the shape space for GPA-aligned specimens has \(\small{2p-4}\) dimensions for 2D data
For 3D data, dimensionality is \(\small{3p-7}\) (general form is: \(\small{pk – k – k(k – 1)/2 – 1}\)
Because these dimensions are redundant, standard parametric statistical hypothesis testing will not work (singular covariance matrix… means divide by zero)
Therefore, the shape space for GPA-aligned specimens has \(\small{2p-4}\) dimensions for 2D data
For 3D data, dimensionality is \(\small{3p-7}\) (general form is: \(\small{pk – k – k(k – 1)/2 – 1}\)
Because these dimensions are redundant, standard parametric statistical hypothesis testing will not work (singular covariance matrix… means divide by zero)
One can eliminate these dimensions via Orthogonal Projection or the thin-plate spline
Now we have: \(\tiny{Y}=\begin{bmatrix} X_1 & Y_1 & Z_1 \\ X_2 & Y_2 & Z_2 \\ \vdots & \vdots & \vdots \\ X_p & Y_p & Z_p \\ \end{bmatrix}\) & \(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)
Now we have: \(\tiny{Y}=\begin{bmatrix} X_1 & Y_1 & Z_1 \\ X_2 & Y_2 & Z_2 \\ \vdots & \vdots & \vdots \\ X_p & Y_p & Z_p \\ \end{bmatrix}\) & \(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)
If shape variation is localized in a few landmarks this can be a problem - The ‘Pinocchio effect’
One may desire methods resistant to this effect
Estimate parameters for translation, scaling and rotation using medians instead of least-squares means
Example: two shapes that differ only in the front triangle
GPA is the preferred method: intuitive criterion for optimization (LS); statistically robust; well known properties of resulting shape space
\[\tiny{D}_{Proc}=\sqrt{\sum_{i,j}^{k,p}\left(\mathbf{Y}_{1.ij}-\mathbf{Y}_{2.ij}\right)^2}\]